Engaging Math Thinking Tasks with a Low Floor and High Ceiling

            In his book, Building Thinking Classrooms in Mathematics, Peter Liljedahl advocates having students work on challenging, highly engaging tasks in mixed-achievement groups of two (at the primary level) or three (in grades 3 to 12). Each group works collaboratively on a vertical whiteboard, standing up, sharing one marker, with the teacher observing their work and, as described in the summary just above, answering only “keep-thinking” questions. Liljedahl suggests starting with fun problems to establish a collaborative culture, then segueing into the regular curriculum, always posing open-ended tasks that challenge students to put their heads together and think at higher levels. The teacher encourages students to observe and learn from other groups’ work and brings closure at the end of each class. From his book, here are some examples of the types of tasks Liljedahl suggests: 

Lower elementary: 

How many squares are in this image? 

             (Image: 4x4 table of squares)

You have 16 jellybeans and four jars: 

    - Place the jellybeans in the jars so that each jar has either 3 or 6 jellybeans. Are there some things that are not possible? 

    - Place the jellybeans such that each jar has one more than the jar before it. How many ways can you do this? 

    - Place the jellybeans so that each jar has twice as many as the jar before it. Three times as many. The 

Ice Dream ice cream shop has 10 flavors of ice cream. How many different two-scoop ice cream cones can you make? What if there were 11 flavors? What if there were 12 flavors? How about with 20 flavors? What if each cone had at most three scoops? 

A farm has some chickens and some pigs. One day the owner notices that the animals have a total of 22 legs. How many chickens and how many pigs might there be? Can you come up with another solution? And another? Can you come up with all the solutions? How do you know that you have all the solutions? 

Upper elementary and middle school: 

If 6 cats can kill 6 rats in 6 minutes, how many will be needed to kill 100 rats in 50 minutes? 

If I were to write the numbers from 1 to 100, how many times would I use the digit 7? What if I wrote 1 to 1000? How many zeroes? 

Select four numbers from 1 to 9 at random. Using these four numbers and any operations, make the values from 1 to 30. 

How many ways are there to make a dollar using only nickels, dimes, and quarters? 

I have a four-minute egg timer and a seven-minute egg timer – the kind you turn over and let the sand run through. Can I use these to cook a nine-minute egg? If so, how long will someone have to wait for their egg? 

High school: 

Decompose 25 using addition, for example: 25 = 10+15, 25 = 10 + 10 + 5… What is the biggest product you can make if you multiply the addends together? 

You want to arrange four candles on a birthday cake. How many ways can you place the candles such that there are no more than two different distances between two candles? 

 From Building Thinking Classrooms in Mathematics by Peter Liljedahl (Corwin, 2021)